1. Field of the Invention
This Invention relates to a method and apparatus for processing electronic signals, and more particularly to matched active filters which have essentially identical average gain.
2. Description of Related Art
In communications systems, information must be encoded in a manner which allows the information to be transmitted from a first location to a second location. For example, transmitters, such as radio stations, commonly transmit FM (frequency modulation) signals. In accordance with frequency modulation techniques, the amplitude of the carrier is essentially constant and the frequency of the carrier varies about the carrier center frequency in proportion to the amplitude of the baseband signal.
In accordance with one method for frequency modulating information upon a carrier, two signals are generated and modulated with the information which is to be communicated. The first such signal (the in-phase or "I component") is identical to the second such signal (the quadrature or "Q component"), with a 90.degree. phase difference between the I and Q components. These two signals are then summed. For example, a baseband signal V(t) is modulated onto a carrier by generating an I and Q component and summing these two signals as follows: EQU V.sub.0 Sin [V(t)]+V.sub.0 Cos[V(t)]
One circuit for generating this modulated carrier is shown in block diagram form in FIG. 1. In accordance with the method implemented by this circuit, the analog baseband signal V(t) is applied to an analog-to-digital (A/D) converter 101. The output from the A/D converter 101 is a digital signal which has been sampled at a predetermined sampling rate which is at least twice the highest frequency expected to be encountered in the baseband signal. Each sample is represented by a digital value which indicates the average amplitude of the baseband signal over the sample time. This digital signal is then applied to a digital signal processor (DSP) 103 which generates the two components: V.sub.0 Sin [V(t)] (referred to as the in-phase component of the carrier) and V.sub.0 Cos [V(t)] (referred to as the quadrature component of the carrier). These two components are then each coupled to a digital-to-analog (D/A) converter 105, 107. The output from each D/A converter 105, 107 is an analog signal which includes quantization noise. Quantization noise is the inaccuracy of the conversions between analog and digital representations which result from the fact that the number of bits in the digital representation of each sample are not infinitely great. This quantization noise is minimized by coupling the output from each D/A converter to an associated conventional analog low pass filter 109, 111, such as an eliptical filter, which has a pair of imaginary zeros at the frequency of the quantization noise.
In addition to the quantization noise, the analog signal which is reconstructed from the samples in the D/A converter will include high frequency components due to the fact that the number of samples is not infinite. Each quantization filter 109, 111 is typically a low pass filter, since the quantization noise and the high frequency component of the analog signal output from the D/A converter will appear as a frequency which is above the sampling rate. In most cases, these analog filters must be fabricated using discrete components. The output from each quantization filter 109, 111 is coupled to a mixer 113, 115 which completes the modulation of the signals and converts the signal to an intermediate frequency (IF). The IF output may then be converted to a radio frequency (RF) signal which is suitable for broadcasting. A local oscillator (LO) 117, provides an LO signal by which the I and Q signals are multiplied in the mixers 113, 115. A 90.degree. phase shifter 120 is provided between the LO 117 and the mixer 115. The output from each mixer 113, 115 is then coupled to a summing circuit 119. The output of the summing circuit 119 is then transmitted.
In accordance with another method for modulating a carrier, digital information is encoded onto the carrier by altering both the amplitude and the phase of the carrier. In accordance with one such method for encoding information referred to as quadrature amplitude modulation (QAM), the carrier is a composite signal which includes a first signal at the carrier frequency (i.e., the I component), and a second signal at the carrier frequency which is shifted in phase 90.degree. from the I component (i.e., the Q component). The relative amplitude of each of the components of the carrier may be either positive or negative and may be any of a predetermined fixed values. Negative amplitudes represent a 180.degree. phase shift from the positive amplitude. Each pair of amplitude values represent a point on a QAM plot, mad also represents a particular digital symbol. FIG. 2 is an illustration of a QAM plot having 16 points which may be represented by each component of the carrier being modulated with one of four amplitude values.
In both the circuit shown in FIG. 1, and a circuit which performs QAM modulation, the amplitude of the I and Q components must be generated without any distortion in the amplitude relationship between the two components. That is, if the amplitude of the I and Q components are processed by components of the system (such as filters) the value of the amplitude of each component is not critical, since the values of the I and Q components may be scaled. However, the relative amplitude of the I component to the Q component must be preserved or the relative position of the symbols which were encoded onto the carrier will be altered. With this in mind, it is clear that in-phase signal filters and quadrature signal filters (such as the low pass filters, each with a notch or imaginary zero) which are commonly used to remove undesirable frequency components from the I and Q components after these components have been generated must have very nearly the same amplitude response as the carrier frequency. It will also be clear to those skilled in the art that a wide variety of other situations exist in which the amplitude relationship between two signals must be preserved after filtering with a low pass filter having a notch.
The requirement that the gain of the quadrature component filter and the in-phase component filter be matched poses a significant challenge to the engineer tasked with designing such filters. Low pass filters having imaginary zeros typically have a frequency response which is commonly described in terms of a transfer function using a parameter "S", which is equal to 2.pi..omega., where .omega. is frequency in radians. A typical expression representing the transfer function of a low pass filter having a notch is: EQU K.multidot.(b.sub.2 .multidot.S.sup.2 +1)/(a.sub.2 .multidot.S.sup.2 +a.sub.1 .multidot.S+1)
It will be understood by those skilled in the art that when b.sub.2 .multidot.S.sup.2 =-1, the transfer function goes to zero (i.e., the value of S for which this is true is a "zero" for that filter). Thus, for any input, the output is zero. This results in a notch at the frequency at which S has a value which satisfies the condition b.sub.2 .multidot.S.sup.2 =-1. When attempting to match two such filters for use in a modulator, the constants, a, b, and K within the first filter must match the corresponding constants within the second filter as closely as possible. The value K is hereafter referred to as the "gain" of the filter. Each of the other constants, a, b, and c control the shape of the filter response. Quite obviously, if the values of each of these constants are identical, then the transfer functions for the two filters will be identical. While it is desirable to equate each these constants, the most critical constant in the transfer function is the gain K, since this constant has an impact on the entire transfer function.
One example of a low pass filter having imaginary zeros is illustrated in the schematic present in FIG. 3. The gain K for the filter of FIG. 3 is: EQU (R.sub.3 .multidot.R.sub.10)/(R.sub.4 .multidot.R.sub.8)-(R.sub.10 /R.sub.9 )
Clearly, the tolerance of each of the components R.sub.3, R.sub.4, R.sub.5, R.sub.8, R.sub.9, and R.sub.10 will control how nearly equal the frequency response of two such filters can be made. Due to the relatively large number of resistors upon which the constant K depends, it would be difficult to provide two filters with transfer functions that are sufficiently similar. That is, each of the resistors in each of the filters would have to be a precision resistance in order to conform that resistor to the corresponding resistor of the other filter and thus match the frequency response of the two filters. Precision resistors are expensive and add to the expense of producing such a pair of filters.
FIG. 3 is a schematic of another filter in which the gain K of the transfer function is far less dependent upon the values of individual components within the filter circuit. However, the gain K is nonetheless dependent upon two resistances, the ratio of which must be equal in the two filters. FIG. 4, is a schematic of yet another example of a filter which might be used in the paths of the in-phase and quadrature components of a demodulator. Once again, the gain K of the transfer function of the filter shown in FIG. 4 is dependent upon the ratio of two resistors.
It should be clear that a number of situations require matched filters, including designing modulators. Therefore, it would be desirable to have matched filters in which the gain of each filter is equal to the gain of the other. The present invention provides such a matched pair of filters.